Abstract
The peculiar band structure of semimetals exhibiting Dirac and Weyl crossings can lead to spectacular electronic properties such as large mobilities accompanied by extremely high magnetoresistance. In particular, two closely neighboring Weyl points of the same chirality are protected from annihilation by structural distortions or defects, thereby significantly reducing the scattering probability between them. Here we present the electronic properties of the transition metal diphosphides, WP_{2} and MoP_{2}, which are typeII Weyl semimetals with robust Weyl points by transport, angle resolved photoemission spectroscopy and first principles calculations. Our single crystals of WP_{2} display an extremely low residual lowtemperature resistivity of 3 nΩ cm accompanied by an enormous and highly anisotropic magnetoresistance above 200 million % at 63 T and 2.5 K. We observe a large suppression of charge carrier backscattering in WP_{2} from transport measurements. These properties are likely a consequence of the novel Weyl fermions expressed in this compound.
Introduction
Recently, many semimetals were found to exhibit Weyl and Dirac crossings in their band structure and, as a consequence high magnetoresistance and high mobilities^{1,2,3}. Dirac points are fourfold degenerate^{4,5,6} whereas Weyl points are twofold degenerate and come in pairs with opposite chirality, namely, a source and a sink of the Berry curvature^{7,8}. In 2012, Na_{3}Bi was the first semimetal predicted to contain Dirac fermions which was soon experimentally verified^{5,9}. The first Weyl semimetal was anticipated^{8} in 2015 and then quickly discovered in TaAs^{10}, and its close relatives^{11,12}. Later, typeII Dirac and Weyl fermions (WSMII) were identified, in which the Dirac or Weyl cones, respectively, are tilted with respect to the Fermi energy^{13,14,15}. The archetypical WSMIIs are the two dimensional van der Waals compounds WTe_{2} and MoTe_{2} ^{13,16,17,18,19}, in which pairs of neighboring Weyl points have opposite chirality. Fermions with even higher degeneracy can be found in compounds with certain symmetries where atoms sit on Wyckoff positions with high multiplicities^{20}. For a compound to display Weyl points it must exhibit either inversion symmetry breaking, as in, for example, TaAs^{8}, or time reversal symmetry breaking, as in, for example, GdPtBi^{21,22}. In these compounds, the Weyl points (WPs) of opposite chirality are close to each other and, hence, are vulnerable to annihilation from structural distortions or defects.
Very recently, the three dimensional transition metal diphosphides, WP_{2} and MoP_{2}, were predicted to host four pairs of typeII WPs below the Fermi energy^{23}. One important characteristic feature of these WSMIIs are that the nearest WPs are of the same chirality and, therefore, are robust against structural distortions or defects. One might then expect high conductivities which indeed we have discovered in WP_{2}. Here we show that WP_{2} exhibits extremely high conductivity and the highest magnetoresistance (MR) values yet observed in any compound with an extraordinarily large mean free path of 0.5 mm. A similar effect is also observed in MoP_{2}.
Results
Structure of tungsten and molybdenum diphosphides
WP_{2} is a three dimensional compound which crystallizes in a nonsymmorphic Cmc2_{1} space group. In this orthorhombic structure, tungsten atoms are surrounded by seven P atoms, six located at the corners of triangular prism and the seventh outside one of the rectangular faces. As can be seen in Fig. 1a the compound contains a mirror plane perpendicular to the aaxis, a cglide perpendicular to the baxis and a twofold screw axis along the caxis. Interestingly, the space group symmetry of WP_{2} is very similar to the two dimensional WTe_{2} which also contains a mirrorplane, a glideplane and a twofold screw axis^{13}. MoP_{2} crystallizes in the same structure. Figure 1b shows a typical crystal of WP_{2} which is needleshaped with its length oriented along the aaxis.
Evolution of Weyl points
In Fig. 1c we depict how a fourfold degenerate Dirac point can be split into two Weyl points with opposite chirality via inversion symmetry breaking (or alternatively in a magnetic field). The protection of these Weyl points against annihilation depends on how far they are separated from each other in momentum space. From abinitio calculations, in the absence of spin orbit coupling (SOC) WP_{2} and MoP_{2} possess two pairs of fourfold degenerate linear band crossing points with opposite sign of Chern numbers ( ± 2). The Chern number is the integral of the Berry curvature around a particular point in momentum space. When SOC is introduced, the spin degeneracy is lifted so that the linear band crossing points evolve into two × twofold degenerate linear band crossing Weyl points of the same chirality. The fact that they have the same chirality means that they are robust (see Fig. 1d). This makes WP_{2} and MoP_{2} unique which differ from other Weyl semimetals and the effect of such robustness of Weyl points can be expected in the electrical transport. We have studied the electrical properties of WP_{2} and MoP_{2} together with theoretical calculations. We focus on the electronic properties of WP_{2} and the data of MoP_{2} are mostly included in the Supplementary Information.
Zero field resistivity behavior
The zero field resistivity of WP_{2} shows a linear temperature dependence at high temperature that is indicative of dominant electronphonon scattering (see Fig. 1e inset). For the several crystals studied (namely, C1–C5, Supplementary Fig. 4), the smallest resistivity observed at 2 K was ρ ~ 3 nΩ cm yielding extremely large residual resistivity ratios (RRRs) = ρ (300 K)/ρ (2 K) with up to RRR ≈ 25,000. To the best of our knowledge, these very high values of the low temperature conductivity and RRR are the largest yet reported in any binary compound. MoP_{2} also shows the similar temperature dependence of resistivity and this reaches from 25.78 μΩ cm at 300 K to 10 nΩ cm at 2 K (see Supplementary Fig. 7a) with RRR = 2578. While this may be an indication of a very high purity of the WP_{2} crystals studied here, another more intriguing possibility is that the unique electronic properties of these compounds make certain scattering mechanisms less likely through topological protection. In order to elucidate this further, we present a detailed temperature dependence of the resistivity at low temperatures in Fig. 1e for WP_{2}. The dependence cannot be accounted for by the usual electronelectron (ee, T ^{2}behavior) and electronphonon scattering (eph, T ^{5}behavior) mechanisms. We observe that the resistivity falls more steeply as the temperature is reduced. One mechanism could be phonon drag, however, phonon drag is usually difficult to observe because it is obscured by electrondefect scattering processes^{24,25}. Phonon drag gives an exponential dependence of ρ (T) which we find considerably improves the fit quality to our data (blue solid line in Fig. 2a). We successfully employ this fitting scheme to other WP_{2} crystals as well (Supplementary Fig. 5). Resistivity of MoP_{2} at 2 K is found to be around one order of magnitude larger than WP_{2}.
Effect of magnetic field on resistivity
Magnetic field is a potent tool to study the motion of electrons inside metals. In general, when the electric and magnetic fields are applied transverse to each other, a positive MR is observed due to the Lorentz force. In WP_{2} despite a very high conductivity, we observe a huge magnetoresistance, as shown in Fig. 2a, which depicts ρ (T) for various magnetic fields applied along the baxis, while the current (I) was applied along the aaxis. The field dependence of the resistivity is quite small for temperatures above 100 K, below which it starts to increase drastically. For fields above 0.5 T we find that ρ (T) displays an upturn below ∼ 50 K which is typical of many semimetals. At 9 T, this amounts to a band gap of 22 meV by fitting the low temperature ρ (T) data by Arrhenius equation. In bismuth and graphite this behavior was argued to be a magnetic fieldinduced excitonicinsulator transition which however requires that the system is near quantum limit^{26,27,28}. Conventional multiband approach has also been undertaken alternatively to explain this transition in compensated semimetals^{29}. The upturn is followed by a peak at lower temperatures which was also seen in bismuth and graphite. It is believed to arise from superconducting correlations when B is larger than the quantum limit^{27}. However, in WP_{2} the peak appears much below the quantum limit which we estimate to be of several thousand tesla. Similar behavior of peak and upturn in ρ (T) data was observed in many other crystals measured (see Supplementary Fig. 6). Further work is required to understand the origin of these effects.
At 2 K and 9 T, WP_{2} exhibits a transverse MR of 4.2 × 10^{6}% (Fig. 2b), which is the largest yet reported in any compound and this retains up to 63 T with a value of 2 × 10^{8}%. We find that the value of MR decreases sharply with decreasing RRR values (Supplementary Fig. 8). An order of magnitude decrease in RRR results in a twoorder of magnitude decrease in the MR value at 2 K and 9 T. Our measured MR is well described by a near parabolic field dependence, MR∝B ^{1.94}, up to the maximum field (63 T) explored as shown in Fig. 2c. Such an accurate scaling of MR with B makes WP_{2} an ideally suited material for accurate magnetic field sensors (only 0.2% error due to quantum oscillations, see Supplementary Fig. 9 and Supplementary Note 2) which can be used in the megagauss regime. Moreover, MoP_{2} also exhibits extremely large parabolic MR with a value of 3.2 × 10^{5}% at 2 K and 9 T (Fig. 2d). The largest MR among several measured crystals of MoP_{2} was 6.5 × 10^{5}% (Supplementary Fig. 7) which is slightly less as compared to WP_{2}.
Fermi surface topology
To understand the remarkable properties of WP_{2} and MoP_{2} further, we have performed electronic band structure calculations based on the density functional theory. The lack of inversion symmetry in WP_{2} leads to a spinsplitting of the bands, and both the electron and hole Fermi surfaces (FSs) come in pairs with Rashbalike splitting, see Supplementary Fig. 1. The hole and electron FSs are located around the X and Y points of the BZ, respectively, as shown in Fig. 3a. The pair of hole FSs are open and spaghettilike extending along the baxis while electrons form a pair of bowtielike closed FSs for WP_{2} and MoP_{2}. From the slope of Hall resistivity vs. B at high magnetic field we obtain dominating holetype carrier concentration and mobility of 5 × 10^{20} cm^{−3} and 4 × 10^{6} cm^{2} V^{−1} s^{−1}, respectively at 2 K, while similar order of carrier density 1.2 × 10^{21} cm^{−3} has also been found at charge neutral point in calculation. Hall resistivity at different temperatures and the corresponding calculated carrier density and mobility of WP_{2} are shown in Supplementary Fig. 10.
Angleresolved photoemission spectroscopy (ARPES)
In order to directly investigate the electronic structure of WP_{2}, we have performed ARPES measurements on the (010) surface with photon energy hv = 50 eV. Both ARPES and theoretical results indicate that no unclosed Fermi arc exists on the (010) surface, since the projections of a pair of Weyl nodes with opposite chirality overlap with each other on the (010) surface. However, we find that the measured Fermi surface and energy dispersion match very closely to the bulk electronic band, verifying the accuracy of our band structure calculations. From AREPES measurements we can see that the FSs in (010) direction contain two types of Fermi surfaces locating around \(\bar {\mathrm{X}}\) and \(\bar \Gamma \)point, respectively, which fit the calculated Femi surface very well (compare Fig. 3a, b). Due to the tubeshape of the hole FSs, their 2D projection along bdirection behaves as a closed loop without any states near the center, as seen around the X point in Fig. 3b. Our calculated electronic band structures are further checked by the comparison with energy dispersions from AREPES in \(\bar{\mathrm A}  \bar{\mathrm X}\) and \(\bar{\mathrm{ Z }}\bar \Gamma \) directions, which cross the hole and electron pockets, respectively. From Fig. 3c, we see that the calculated energy dispersion for the valence bands fit the ARPES measurements very well in the \(\bar{\mathrm A}  \bar{\mathrm X}\) direction in a large energy window of −1.4–0 eV. The energy dispersion in the \(\bar{\mathrm Z }  \bar \Gamma \) direction contains both valence and conductions bands, as shown in Fig. 3d. Because of the small photon energy involved in the ARPES measurements, some bulk states are not observed, but for all the measured states we can find the correspondence from the calculations. Further studies on other surfaces, especially on the (001) surface, are needed to identify the possible arc states.
Anisotropy in transport
The angular dependence of MR of a compound is a direct reflection of the FS topology. A magnetic field, B baxis will lock the charge carriers with a cyclotron motion around the FSs perpendicular to baxis and a large MR is expected. While tuning the direction of magnetic field from b to caxis, the perpendicular cross section area of the FS changes smoothly and becomes infinite when the field is parallel to caxis owing to the shape of spaghettitype open FSs, and would lead to a dramatic drop of MR, which is consistent with the measured anisotropic MR (Fig. 3e). By contrast, the bowtielike electron FSs are closed pockets with smaller anisotropies perpendicular to aaxis. Thus, the anisotropic MR is mainly due to the hole FSs. Moreover, the shape of the FSs are robust over a large energy range from −0.1 to 0.1 eV, which would lead to an insensitivity of the large MR to doping. The anisotropy of the MR shown in Fig. 3b was measured by rotating the WP_{2} crystal around the aaxis with a magnetic of 9 T varied within the bcplane. The current was applied along the aaxis. The MR is maximum when the field is along the baxis (0^{o}) and decreases by 2.5 orders of magnitude when the field is oriented along the caxis (90^{o}) (Fig. 3b). Such a large anisotropy in MR is rare in a 3D compound and typically seen in 2D van der Waals compounds. Surprisingly, the effect is much more pronounced compared to 2D WTe_{2} (Supplementary Fig. 11). Large anisotropy in MR is also observed in MoP_{2} (Supplementary Fig. 12).
Quantum oscillations
The extremal cross section area of the Fermi surface perpendicular to the applied magnetic field is directly related to the frequency of the quantum oscillations. To map the FS experimentally, we have employed resistivity measurements of WP_{2} in static magnetic field of 33 T along baxis and pulsed fields of 65 T along baxis for MoP_{2}. Figure 4a shows the resistivity of WP_{2} at different temperatures between 2 and 6 K. The Shubnikov–de Haas oscillations (SdH) are clearly visible by subtracting a cubic polynomial from the resistivity data. The extracted amplitudes of the SdH oscillations in WP_{2} as a function of the inverse magnetic field are shown in Fig. 4b. The fast Fourier transform (FFT) of the SdH oscillations identifies four fundamental frequencies which are identified from the spaghettitype holes FS (α′ at 1460 T and α″ at 1950 T) and from the bowtietype electrons FS (β′ at 2650 T and β″ at 3790 T) with the help of abinitio calculations. The quantum oscillations were calculated from the kspace areas of the extremal crosssections of the FSs with a magnetic field along y. We found four frequencies at ~ 1300 T, 1900 T from 2 hole like FSs, and 2800 and 3900 T from 2 electron like FSs, which fit the experimental results well, as presented in Fig. 4c. We calculate the effective mass (m*) of the electrons in α′ and α″ pockets from the temperature dependence of the SdH amplitudes (Fig. 4d) using the Lifshitz–Kosevich (LK) formula: ΔR = X/sinh(X), where X = 14.69 m ^{*} T/B and B is the average field. m* for holes in α′ and α″ pockets are 1.67m _{0} and 1.89m _{0}, respectively. For the electrons in β′ pocket, m* is 1.32m _{0} (see Supplementary Fig. 13), however we could not obtain m* of β″ because of the small amplitude associated with it. If we consider circular Fermi surface cross section of α′ hole band along the baxis which is a fair approximation to make, we can calculate the Fermi area A _{ F } from the Onsanger relation: \(F = (\hbar /{2\pi e}) A_F\) to be 0.14 Å^{−2}. This gives rise to the Fermi vector k _{ F } of 0.21 Å^{−1} and matches quite well to the Fermi cross section of this band normal to the baxis as observed in the ARPES measurements of WP_{2}. A very large Fermi velocity of 1.4 × 10^{5} m/s is obtained from the relation, \(v_F = \frac{{\hbar k_F}}{{m^*}}\). Similarly, for MoP_{2}, we employ pulsed magnetic field up to 63 T to study the SdH oscillations which agree well with the calculated Femi surfaces (Supplementary Fig. 14).
Discussion
We now consider the origin of large conductivity and MR in WP_{2}. The Weyl point induced spin texture (Berry phase) (Supplementary Note 1 and Supplementary Figs. 1 and 2) can effectively suppress the backscattering. The robustness of these Weyl points in WP_{2} due to same chirality of the neighboring Weyl nodes will enhance such effect. The experimental proof of the suppression comes from the ratio, r of transport lifetime (τ _{tr}) and quantum lifetime (τ _{q}) of scattering. τ _{tr} is calculated from the Drude model as \(\tau _{tr} = {\mu m^*}/ e = 3.8 \times 10^{  9}\)s, where μ is the mobility and m* is the effective mass of α′ band. This also gives rise an extraordinarily large classical mean free path of 0.5 mm. τ _{q} is obtained from the broadening of the SdH oscillations as \(\tau _q = \hbar / \left( {2\pi T_{\rm D}} \right)\), where T _{D} is the Dingle temperature. With T _{D} of 1.54 K for α band we obtain τ _{q} = 7.9 × 10^{−13} s. The ratio r = 5000 thus indicates the large suppression of the backscattering of carriers which is comparable to Cd_{3}As_{2} ^{1}. The large value of r also indicates the fact that the momentum conserving processes (electron–electron scattering and phonon drag) are in balance with the momentum relaxing processes (electrondefect, electron phonon, Umklapp scatterings) making WP_{2} a good candidate for observing hydrodynamic flow of electrons. In fact, we have observed a clear signature of hydrodynamic flow in WP_{2} by undertaking size dependent transport measurements^{30}. Therefore, hydrodynamic effects can play significant role in the large conductivity in WP_{2}. Moreover, we cannot also rule out the effect of SOC induced spin splitting. Interestingly, MoP_{2} with smaller SOC, exhibits one order of less conductivity compared to WP_{2}. The large conductivity in WP_{2} at low temperature ensures big RRR value. The unusually large value of RRR has a significant role towards the enhancement of MR. Recently, in Dirac semimetal PtBi_{2}, it was shown that the large RRR value is one of the main factors for large MR ^{31}. Carrier compensation in semimetals also gives rise to large parabolic MR. Our first principles calculations predict equisized electron and hole pockets, which was also confirmed by Fermi surface obtained from the ARPES measurements. In order to further verify, we fit our low temperature Hall conductivity to two band model (see Supplementary Fig. 10 for details). This provides a near compensation of holes (1.5 × 10^{20} cm^{−3}) and electrons (1.4 × 10^{20} cm^{−3}) at 2 K. Hence, the large mobility, extremely large RRR, charge compensation all contribute to the ultrahigh nonsaturating parabolic MR in WP_{2}.
Having seen the extremely large MR and conductivity in WP_{2} and MoP_{2}, we compare these quantities with several topological metals and other wellknown and highly conducting trivial metals in Fig. 5. WP_{2} and MoP_{2} perform much better than Dirac semimetal Cd_{3}As_{2} ^{1} which also exhibits large MR and conductivity. Other semimetals like NbP, WTe_{2}, TaAs, and so on^{2,32} where the MR is quite large, conductivity is orders of magnitudes smaller because of small carrier concentrations. In copper, which is one of the most conductive metals known, the MR is small and is of the order of 50–250 % in single crystals with RRR = 40,000–62,000^{33}. Another class of highly conducting materials with very large RRR are the rutile and delafossite oxides such as IrO_{2} ^{34} and PdCoO_{2} ^{35}. Here the MR is very low compared to semimetals, which typically, however, have low conductivities due to their small carrier concentrations, for example, NbP^{2} and NbSb_{2} ^{36}. In conclusion, WP_{2} and MoP_{2} have conductivities comparable to those in metals like copper while still exhibiting MR values more than any Dirac or Weyl semimetals known.
Although, chiral pumping of charge between the Weyl nodes of opposite chiralities (chiral anomaly) are possible in WSMIIs, it is much more difficult to detect this in WSMIIs compared to standard Weyl semimetals because it can only be observed when parallel electric and magnetic fields are applied along certain crystal directions. We do not observe any negative MR when we apply B and I along aaxis in WP_{2} (see Supplementary Fig. 15). WP_{2} is predicted to exhibit the effect of chiral anomaly only when both electric and magnetic fields are applied along baxis^{23}. The asgrown crystals of WP_{2} are all needleshaped with their length aligned along aaxis, which, therefore, makes it very difficult to apply the electric field along baxis. Another difficulty to observe the chiral anomaly in WP_{2} is its extremely large positive MR when the field is aligned along baxis. A slight disorientation of the field from the baxis results in a large positive MR which would make the observation of the chiral anomaly even more difficult. We believe that these limitations can be overcome by using a focussed ion beam to fabricate a better sample.
In conclusion, we have shown that WP_{2} is a remarkable compound with properties unlike any other compound yet studied in the families of Dirac, Weyl and novel fermion materials. It displays record breaking RRR values and ultrahigh low temperature conductivities and a nonsaturating magnetoresistance. One of the most interesting questions is the degree to which the topological electronic properties of this material account for its unusual properties. We observe a large suppression of backscattering of electrons and, considering the fact that no special procedures were used to purify the elemental starting materials, we conjecture that the protection of the Weyl points from annihilation plays an important role. This will be an important focus of future work.
Methods
Single crystals growth
Crystals of WP_{2} were prepared by chemical vapor transport method. The single crystals of WP_{2} and MoP_{2} were grown by chemical vapor transport. Starting materials were red phosphorous (AlfaAesar, 99.999%) and tungsten/molybdenum trioxide (AlfaAesar, 99.998%) with iodine as a transport agent. The materials were taken in an evacuated fused silica ampoule. The transport reaction was carried out in a twozonefurnace with a temperature gradient of 1000 °C (T1) to 900 °C (T2) for serval weeks^{37}. After reaction, the ampoule was removed from the furnace and quenched in water. The metallicneedle crystals were characterized by Xray diffraction (see Supplementary Fig. 3).
Electrical transport measurements
Resistivity measurements were performed in a physical property measurement system (PPMS9T, Quantum Design) using the ACT and Resistivity with rotator option. For longitudinal resistivity, linear contacts were made on the naturally grown crystals by silver paint and 25 μm platinum wires. The longitudinal and Hall resistivity were measured in 4wires and 5wires geometry, respectively using a current of 3.0–5.0 mA at temperature range from 2 to 300 K and magnetic fields up to 9 T.
4point resistivity measurements at high static magnetic field were performed at HFML, Nijmegen, Netherlands. The sample was placed on a commercially supplied chip carrier (insulated using a layer of cigarette paper). 25 μm gold wire and 4929 silver paste were used to make contacts between the chip carrier and the sample. An AC current of 1 mA (using a Keithley 6221 current source) was applied along the aaxis, and the voltage was measured along the same direction using a Stanford Research SR 830 lockin amplifier at a frequency of 13 Hz. The sample temperature was controlled by a 4He flowcryostat, and applied fields up to 33 T were generated using a resistive Bitter magnet available at the HFML. The high pulsed fielddependent resistivity was measured in a four point geometry using a 62 T nondestructive pulsed magnet driven by a capacitor bank at the Dresden High Magnetic Field Laboratory. The excitation current was 1 mA with a frequency of 3333 and 7407 kHz.
ARPES measurements
ARPES measurements were performed with VGScienta R4000 electron analyzers at SIS beamline at Swiss Light Source, Paul Scherrer Institut. The energy and angular resolutions were set at 15 meV and 0.2°, respectively. Samples were cleaved insitu along the (010) crystal plane in an ultrahigh vacuum of 5 × 10^{−11} Torr. A shiny mirrorlike surface was obtained after cleaving the samples, confirming their high quality. The Fermi level of the samples was referenced to that of a gold film evaporated onto the sample holder.
Band structure calculations
The electronic structures were calculated by the abinitio calculations based on the density functional theory. We have used the projected augmented wave method as implemented in the program of Vienna abinitio Simulation Package (VASP)^{38}. For getting accurate band structures the exchange and correlation energy was considered in the modified BeckeJohnson (MBJ) exchange potential^{39,40}. Fermi surfaces were interpolated in a dense kgrids of 500 × 500 × 500 points by using maximally localized Wannier functions^{40}.
Data availability
The data that support the findings of this study are available from the corresponding authors N.K. and C.F. upon request.
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Acknowledgements
This work was financially supported by the ERC Advanced Grant No. (742068) “TOPMAT”. We acknowledge Prof. Stuart Parkin for fruitful discussions. We acknowledge the support of the High Field Magnet Laboratory Nijmegen (HFMLRU/FOM), and High Magnetic Field Laboratory Dresden (HLD) at HZDR members of the European Magnetic Field Laboratory (EMFL). The ARPES studies in the work was supported by NCCRMARVEL funded by the Swiss National Science Foundation.
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N.K., C.S., and C.F. designed the experiments. N.K. and C.S. performed transport measurements. V.S. and M.S. grew single crystals. K.M. and H.B. performed Laue xray diffraction experiments. I.L., O.Y., U.Z., and T.F. performed high magnetic field transport measurements. Y.S. with inputs from B.Y. carried out theoretical calculations. N.X., M.Y., and M.S. performed ARPES measurements. N.K., C.S., and C.F. wrote the manuscript with inputs from all the authors.
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Correspondence to Nitesh Kumar.
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Kumar, N., Sun, Y., Xu, N. et al. Extremely high magnetoresistance and conductivity in the typeII Weyl semimetals WP_{2} and MoP_{2} . Nat Commun 8, 1642 (2017). https://doi.org/10.1038/s4146701701758z
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